Attractors for non-compact semigroups via energy equations

The energy equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three speci c and physically relevant problems are considered, namely the ows of a second-grade uid, the ows of a newtonian uid in an in nite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.